Step |
Hyp |
Ref |
Expression |
1 |
|
reccl |
|- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) e. CC ) |
2 |
|
recne0 |
|- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) =/= 0 ) |
3 |
|
eflog |
|- ( ( ( 1 / A ) e. CC /\ ( 1 / A ) =/= 0 ) -> ( exp ` ( log ` ( 1 / A ) ) ) = ( 1 / A ) ) |
4 |
1 2 3
|
syl2anc |
|- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` ( 1 / A ) ) ) = ( 1 / A ) ) |
5 |
4
|
eqcomd |
|- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) = ( exp ` ( log ` ( 1 / A ) ) ) ) |
6 |
5
|
oveq2d |
|- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / ( 1 / A ) ) = ( 1 / ( exp ` ( log ` ( 1 / A ) ) ) ) ) |
7 |
|
eflog |
|- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = A ) |
8 |
|
recrec |
|- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / ( 1 / A ) ) = A ) |
9 |
7 8
|
eqtr4d |
|- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = ( 1 / ( 1 / A ) ) ) |
10 |
1 2
|
logcld |
|- ( ( A e. CC /\ A =/= 0 ) -> ( log ` ( 1 / A ) ) e. CC ) |
11 |
|
efneg |
|- ( ( log ` ( 1 / A ) ) e. CC -> ( exp ` -u ( log ` ( 1 / A ) ) ) = ( 1 / ( exp ` ( log ` ( 1 / A ) ) ) ) ) |
12 |
10 11
|
syl |
|- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` -u ( log ` ( 1 / A ) ) ) = ( 1 / ( exp ` ( log ` ( 1 / A ) ) ) ) ) |
13 |
6 9 12
|
3eqtr4d |
|- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = ( exp ` -u ( log ` ( 1 / A ) ) ) ) |
14 |
13
|
3adant3 |
|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= _pi ) -> ( exp ` ( log ` A ) ) = ( exp ` -u ( log ` ( 1 / A ) ) ) ) |
15 |
14
|
fveq2d |
|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= _pi ) -> ( log ` ( exp ` ( log ` A ) ) ) = ( log ` ( exp ` -u ( log ` ( 1 / A ) ) ) ) ) |
16 |
|
logrncl |
|- ( ( A e. CC /\ A =/= 0 ) -> ( log ` A ) e. ran log ) |
17 |
16
|
3adant3 |
|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= _pi ) -> ( log ` A ) e. ran log ) |
18 |
|
logef |
|- ( ( log ` A ) e. ran log -> ( log ` ( exp ` ( log ` A ) ) ) = ( log ` A ) ) |
19 |
17 18
|
syl |
|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= _pi ) -> ( log ` ( exp ` ( log ` A ) ) ) = ( log ` A ) ) |
20 |
|
df-ne |
|- ( ( Im ` ( log ` A ) ) =/= _pi <-> -. ( Im ` ( log ` A ) ) = _pi ) |
21 |
|
lognegb |
|- ( ( ( 1 / A ) e. CC /\ ( 1 / A ) =/= 0 ) -> ( -u ( 1 / A ) e. RR+ <-> ( Im ` ( log ` ( 1 / A ) ) ) = _pi ) ) |
22 |
1 2 21
|
syl2anc |
|- ( ( A e. CC /\ A =/= 0 ) -> ( -u ( 1 / A ) e. RR+ <-> ( Im ` ( log ` ( 1 / A ) ) ) = _pi ) ) |
23 |
22
|
biimprd |
|- ( ( A e. CC /\ A =/= 0 ) -> ( ( Im ` ( log ` ( 1 / A ) ) ) = _pi -> -u ( 1 / A ) e. RR+ ) ) |
24 |
|
ax-1cn |
|- 1 e. CC |
25 |
|
divneg2 |
|- ( ( 1 e. CC /\ A e. CC /\ A =/= 0 ) -> -u ( 1 / A ) = ( 1 / -u A ) ) |
26 |
24 25
|
mp3an1 |
|- ( ( A e. CC /\ A =/= 0 ) -> -u ( 1 / A ) = ( 1 / -u A ) ) |
27 |
26
|
eleq1d |
|- ( ( A e. CC /\ A =/= 0 ) -> ( -u ( 1 / A ) e. RR+ <-> ( 1 / -u A ) e. RR+ ) ) |
28 |
23 27
|
sylibd |
|- ( ( A e. CC /\ A =/= 0 ) -> ( ( Im ` ( log ` ( 1 / A ) ) ) = _pi -> ( 1 / -u A ) e. RR+ ) ) |
29 |
|
negcl |
|- ( A e. CC -> -u A e. CC ) |
30 |
|
negeq0 |
|- ( A e. CC -> ( A = 0 <-> -u A = 0 ) ) |
31 |
30
|
necon3bid |
|- ( A e. CC -> ( A =/= 0 <-> -u A =/= 0 ) ) |
32 |
31
|
biimpa |
|- ( ( A e. CC /\ A =/= 0 ) -> -u A =/= 0 ) |
33 |
|
rpreccl |
|- ( ( 1 / -u A ) e. RR+ -> ( 1 / ( 1 / -u A ) ) e. RR+ ) |
34 |
|
recrec |
|- ( ( -u A e. CC /\ -u A =/= 0 ) -> ( 1 / ( 1 / -u A ) ) = -u A ) |
35 |
34
|
eleq1d |
|- ( ( -u A e. CC /\ -u A =/= 0 ) -> ( ( 1 / ( 1 / -u A ) ) e. RR+ <-> -u A e. RR+ ) ) |
36 |
33 35
|
syl5ib |
|- ( ( -u A e. CC /\ -u A =/= 0 ) -> ( ( 1 / -u A ) e. RR+ -> -u A e. RR+ ) ) |
37 |
29 32 36
|
syl2an2r |
|- ( ( A e. CC /\ A =/= 0 ) -> ( ( 1 / -u A ) e. RR+ -> -u A e. RR+ ) ) |
38 |
28 37
|
syld |
|- ( ( A e. CC /\ A =/= 0 ) -> ( ( Im ` ( log ` ( 1 / A ) ) ) = _pi -> -u A e. RR+ ) ) |
39 |
|
lognegb |
|- ( ( A e. CC /\ A =/= 0 ) -> ( -u A e. RR+ <-> ( Im ` ( log ` A ) ) = _pi ) ) |
40 |
38 39
|
sylibd |
|- ( ( A e. CC /\ A =/= 0 ) -> ( ( Im ` ( log ` ( 1 / A ) ) ) = _pi -> ( Im ` ( log ` A ) ) = _pi ) ) |
41 |
40
|
con3d |
|- ( ( A e. CC /\ A =/= 0 ) -> ( -. ( Im ` ( log ` A ) ) = _pi -> -. ( Im ` ( log ` ( 1 / A ) ) ) = _pi ) ) |
42 |
41
|
3impia |
|- ( ( A e. CC /\ A =/= 0 /\ -. ( Im ` ( log ` A ) ) = _pi ) -> -. ( Im ` ( log ` ( 1 / A ) ) ) = _pi ) |
43 |
20 42
|
syl3an3b |
|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= _pi ) -> -. ( Im ` ( log ` ( 1 / A ) ) ) = _pi ) |
44 |
|
logrncl |
|- ( ( ( 1 / A ) e. CC /\ ( 1 / A ) =/= 0 ) -> ( log ` ( 1 / A ) ) e. ran log ) |
45 |
1 2 44
|
syl2anc |
|- ( ( A e. CC /\ A =/= 0 ) -> ( log ` ( 1 / A ) ) e. ran log ) |
46 |
|
logreclem |
|- ( ( ( log ` ( 1 / A ) ) e. ran log /\ -. ( Im ` ( log ` ( 1 / A ) ) ) = _pi ) -> -u ( log ` ( 1 / A ) ) e. ran log ) |
47 |
45 46
|
stoic3 |
|- ( ( A e. CC /\ A =/= 0 /\ -. ( Im ` ( log ` ( 1 / A ) ) ) = _pi ) -> -u ( log ` ( 1 / A ) ) e. ran log ) |
48 |
43 47
|
syld3an3 |
|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= _pi ) -> -u ( log ` ( 1 / A ) ) e. ran log ) |
49 |
|
logef |
|- ( -u ( log ` ( 1 / A ) ) e. ran log -> ( log ` ( exp ` -u ( log ` ( 1 / A ) ) ) ) = -u ( log ` ( 1 / A ) ) ) |
50 |
48 49
|
syl |
|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= _pi ) -> ( log ` ( exp ` -u ( log ` ( 1 / A ) ) ) ) = -u ( log ` ( 1 / A ) ) ) |
51 |
15 19 50
|
3eqtr3d |
|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= _pi ) -> ( log ` A ) = -u ( log ` ( 1 / A ) ) ) |